Monopoles and Modif ications of Bundles over Elliptic Curves
?Andrey M. LEVIN †‡, Mikhail A. OLSHANETSKY †§ and Andrei V. ZOTOV †§
† Max Planck Institute of Mathematics, Bonn, Germany
‡ Institute of Oceanology, Moscow, Russia E-mail: alevin@wave.sio.rssi.ru
§ Institute of Theoretical and Experimental Physics, Moscow, Russia E-mail: olshanet@itep.ru, zotov@itep.ru
Received November 20, 2008, in final form June 10, 2009; Published online June 25, 2009 doi:10.3842/SIGMA.2009.065
Abstract. Modifications of bundles over complex curves is an operation that allows one to construct a new bundle from a given one. Modifications can change a topological type of bundle. We describe the topological type in terms of the characteristic classes of the bundle.
Being applied to the Higgs bundles modifications establish an equivalence between different classical integrable systems. Following Kapustin and Witten we define the modifications in terms of monopole solutions of the Bogomolny equation. We find the Dirac monopole solution in the caseR×(elliptic curve). This solution is a three-dimensional generalization of the Kronecker series. We give two representations for this solution and derive a functional equation for it generalizing the Kronecker results. We use it to define Abelian modifications for bundles of arbitrary rank. We also describe non-Abelian modifications in terms of theta- functions with characteristic.
Key words: integrable systems; field theory; characteristic classes
2000 Mathematics Subject Classification: 14H70; 14F05; 33E05; 37K20; 81R12
1 Introduction
The modifications (or the Hecke transformation) of bundles over complex curves is a corre- spondence between two bundles E and ˜E. It is isomorphism in a complement of some divisor.
A modification can change the topological type of the original bundle. From the field-theoretical point of view the modification is provided by a gauge transformation of sections, which is sin- gular at the divisor. In [1] we apply this procedure to the Higgs bundles. The Higgs bundles are the phase spaces of the Hitchin integrable systems [2]. Modifications acts on the phase space as a symplectic transformation. In this special case we call the modification the Symplectic Hecke Correspondence. For the Higgs bundles over elliptic curves with marked points Symplec- tic Hecke Correspondence leads to a symplectomorphism between different classical integrable systems such as
• Elliptic Calogero–Moser system ⇔Elliptic GL(N,C) Top, [1];
• Calogero–Moser field theory⇔ Landau–Lifshitz equation, [1,4];
• Painlev´e VI⇔ non-autonomous Zhukovsky–Volterra gyrostat, [3].
?This paper is a contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions” (July 21–25, 2008, MPIM, Bonn, Germany). The full collection is available athttp://www.emis.de/journals/SIGMA/Elliptic-Integrable-Systems.html
In these examples modifications increase the degree of the underlying bundles on one. In general, modifications act as the B¨acklund transformations of integrable systems. If degree of the bundles (modula rank) is not changed then modifications produce what is called the autoB¨acklund transformations. It turned out that the modification in the first example is equivalent to the twist of R-matrices [5, 6] that transforms the dynamical R-matrices of the IRF models of the GL(N) type [7] to the vertex R-matrices [8] corresponding to the GL(N) generalization of the XYZ models.
The modifications are parameterized by vectors m~ of the weight lattices P of SL(N,C).
If m~ belongs to the root sublattice Q ⊂ P, then the modified bundle ˜E has the same degree as E. Otherwise, the degree of bundle is changed. The modifications can be described by changing another topological invariant. It is a characteristic class of a bundle. Let the base of E be a Riemann surface Σg of genus g. Then the characteristic class of E is an element of H2(Σg,ZN) ∼ ZN, where ZN ∼ P/Q is a center of SL(N,C). Another example of the characteristic classes, is the characteristic class of spin-bundles, that will not considered here, is the Stiefel–Whitney classH2(Σg, Z2).
Here we discuss a field-theoretical interpretation of modifications. It was established in [9]
that the modifications are related to the Dirac monopole configurations in a topological version of the N = 4 four-dimensional super-symmetric Yang–Mills theory. If “the space-time” of the topological theory has the formR2×Σg, then the modifications ofE over Σg are parameterized by the monopoles charges.
To describe the modification it is sufficient to neglect the “time” dependence and consider R×Σg. The condition for fields to preserve the supersymmetry amounts to the Bogomolny equation.
The aims of this paper are
• To define modifications and describe their interrelations with the Bogomolny equation following [9]. We consider a special configuration of the space-timeR2×Στ, where Στ is an elliptic curve with the modular parameter τ.
• To find solutions of the Bogomolny equation in the case of line bundles over Στ. They are generalizations of the Kronecker series [17]. We give two representations of the solution and prove their equivalence by means of the functional equation generalizing the Kronecker functional equation.
• To describe non-Abelian modifications that are not related directly to solutions of the Bogomolny equation and follows from our previous results.
2 Characteristic classes of holomorphic bundles over complex curves
We describe holomorphic bundles over complex curves Σg of genusgand define their character- istic classes.
2.1 Global description
Let π1(Σg) be a fundamental group of Σg. It has 2g generators {aα, bα}, corresponding to the fundamental cycles of Σg with the relation
g
Y
α=1
[aα, bα] = 1, (2.1)
where [aα, bα] =aαbαa−1α b−1α is the group commutator.
Letρ be a representation of π1 in CN. Consider a holomorphic adjoint GL(N,C) bundle E over Σg. In fact,E is a PGL(N,C)∼PSL(N,C) bundle, because the center of GL(N,C) does not act in the adjoint representation. The bundle E can be defined by holomorphic transition matrices of its sections s∈Γ(E) around the fundamental cycles. Let z ∈Σg be a fixed point.
Then
s(aαz) =ρ(aα)s(z), s(bβz) =ρ(bβ)s(z).
Due to (2.1) we have
g
Y
α=1
[ρ(aα), ρ(bα)] = Id. (2.2)
LetK be an extension ofπ1 by the cyclic groupZN ∼Z/NZ
1→ZN → K →π1(Σg)→1. (2.3)
The groupK is defined by the relation
g
Y
α=1
[aα, bα] =ω, ωN = 1.
Let ˆρ be a representation of K in GL(N,C). Then using ˆρ as transition matrices we define a bundle over Σg. But now instead of (2.2) we have
g
Y
α=1
[ ˆρ(aα),ρ(bˆ α)] =ωId. (2.4)
Here ωId is the generator of the center Z(SL(N,C)) ∼ ZN of SL(N,C). It means that ˆρ can serve as transition matrices only for PSL(N,C) bundles, but not for SL(N,C) or GL(N,C) bundles. Note, that the fibers of the PSL(N,C)-bundles are spaces of representations with highest weights from the root lattice Q (A.2) including the adjoint representation with the highest weight $1+$N−1 (A.3). For the SL(N,C) representations the highest weights belong to the weight lattice P (A.4). In this way elements from the factor group P/Q∼ Z(SL(N,C)) (A.6) define an obstruction to lift PSL(N,C) bundles to SL(N,C) bundles.
The obstruction has a cohomological interpretation. Consider the exact sequence following from (A.1)
→H1(Σg,SL(N,C))→H1(Σg,PSL(N,C))→H2(Σg,Z(SL(N,C)))→ · · · .
The groups H1(Σg,SL(N,C)), H1(Σg,PSL(N,C)) are the moduli space of SL(N,C) and PSL(N,C) bundles. Then H2(Σg,Z(SL(N,C))) defines an obstruction to lift PSL(N,C) bund- les to SL(N,C) bundles. We callξ∈H2(Σg,ZN)the characteristic class of a PSL(N,C) bundle.
In fact, H2(Σg,ZN)∼ZN and ω in (2.4) represents ξ ∈H2(Σg,ZN).
This construction can be generalized to any factor-group Gl = SL(N,C)/Zl, where l is a nontrivial divisor of N, (N = pl, l 6= 1, N). Consider an extension Kl of π1(Σg) by Zl
(compare with (2.3))
1→Zl→ Kl →π1(Σg)→1.
LetElbe a holomorphicGl-bundle. The fibers ofElbelong to a irreducible representation ofGl
with a highest weightν ∈Γ(Gl) (A.7). Then the transition matrices representing Kl satisfy the relation
g
Y
α=1
[ ˆρ(aα),ρ(bˆ α)] =ωpId, (ωp)l= 1. (2.5)
It follows from the exact sequence 1→Zl→SL(N,C)→Gl →1,
that elements from H2(Σg,Zl)∼Zl are obstructions to lift Gl bundle El to a SL(N,C)-bundle.
The group Zl can be identified with the center of the dual group LGl ∼ Gp = SL(N,C)/Zp
(see (A.11) and (A.10)). Thus, the obstructions to liftGl bundlesEl to a SL(N,C) bundles are defined by H2(Σg,Z(LGl)).
On the other hand, sinceZp is a center ofGl we have the sequence 1→Zp→Gl→PSL(N,C)→1,
where Zp is a center of Gl. Then elements from H2(Σg,Z(Gl)) are obstructions to lift a PSL(N,C)-bundle to aGl-bundle. Summarizing we have defined two types of the characteristic classes
H2(Σg,Z(Gl))−obstructions to lift a PSL(N,C) bundle to aGl bundle,
H2(Σg,Z(LGl))−obstructions to lift a Gl bundle to a SL(N,C) bundle. (2.6) Though forω 6= 1 PSL(N,C) bundles cannot be lifted to SL(N,C) bundles, they can be lifted to GL(N,C) bundles. From the exact sequence
1→ O∗ det→GL(N,C)→PGL(N,C)→1 we have
H1(Σg,GL(N,C))→H1(Σg,PGL(N,C))→H2(Σg,O∗).
The Brauer groupH2(Σg,O∗) vanishes and therefore, there is no obstruction to lift PGL(N,C)∼ PSL(N,C) bundles to GL(N,C) bundles. We will demonstrate it below.
2.2 Holomorphic bundles over elliptic curves
We define an elliptic curve (g = 1) as the quotient Στ = C/(Z+τZ). In this case we can construct explicitly the generic transition matrices for Gl-bundles.
The curve has two fundamental cyclesa: (z→ z+ 1), b : (z →z+τ). We define a trivial bundle E over Στ by two commuting matrices
s(z+ 1) =ρas(z), s(z+τ) =ρbs(z), [ρa, ρb] = Id. (2.7) It is a PGL(N,C)-bundle that can be lifted to SL(N,C) bundles.
Consider a representation of ˆρ of K acting on the sections of E as s(z+ 1) = ˆρas(z), s(z+τ) = ˆρbs(z).
with commutation relation (2.4) [ ˆρa,ρˆb] =ωId.
One can choose
ˆ
ρa=Q, ρˆb= Λ, Q= diag 1, ω, . . . , ωN−1
, Λ =
0 1 . . . 0
0 0 1 0
... ... . .. 1 1 0 . . . 0
. (2.8)
The bundle with these transition functions cannot be lifted to SL(N,C) bundles. Replace ˆρb by ˆ
ρ0b= exp
−2πi N
z+τ
2
Λ. (2.9)
It is a GL(N,C) bundle since [ ˆρa,ρˆ0b] =Id and det ˆρ0b 6= 1 . It follows from (2.9) that a section of the determinant bundle is the theta-function
ϑ(z, τ) =q18 X
n∈Z
(−1)neπi(n(n+1)τ+2nz), q= exp 2πiτ. (2.10)
It has a simple pole in the fundamental domainC/(Z⊕τZ). Therefore, the bundle has degree one. It is called the theta-bundle.
To consider a general case [10] represent the rank as the productN =pl. Define the transition matrix
ˆ
ρa=Q, (2.11)
ˆ
ρb=e(~ul)Λp, (2.12)
where
~
ul= diag(
l
z }| {
up, . . . ,up), up = (˜u1, . . . ,u˜p).
Since [Q,Λp] =ωpIdl,ωp = exp2πil [ ˆρa,ρˆb] =ωpIdN.
Comparing this relation with (2.5) we conclude that (2.11) and (2.12) serve as the transition ma- trices for a Gl-bundle over Στ. Thereforeωp represents an element from H2(Στ,Z(LGl))∼Zp. It is an obstruction (2.6).
As in (2.9), modify the transition matrix ˆ
ρb→ρˆ0b= exp
−2πi p
z+τ
2
ˆ ρb.
We come to the GL(N,C)-bundle of degreep (modN).
2.3 Local description
There exists another description of a holomorphic bundles over Σg. Let w0 be a fixed point on Σg and Dw0 (D×w0) be a disc (punctured disc) with a center w0 with a local coordinate z.
A bundle E over Σg can be trivialized over D and over Σg \w0. These two trivializations are related by a GL(N,C) transformation g(z), holomorphic on D×w0. If we consider another trivialization over D then g is multiplied from left by an invertible matrix h on D. Likewise, a trivialization over Σg\w0 is determined up to the multiplication on the rightg→gh, where h∈GL(N,C) is holomorphic on Σg\w0. Thus, the set of isomorphism classes of rankN vector bundles is described as a double-coset
GL(N,C)(Dw0)\GL(N,C)(D×w0)/GL(N,C)(Σg\w0),
where GL(N,C)(U) denote the group of GL(N,C)-valued holomorphic functions onU.
Let detg(z) = 1. If g(ze2πi) =g(z) then it defines a SL(N,C)-bundle over Σg. But if the monodromy is nontrivial
g(ze2πi) =ωg(z), ωN = 1, (2.13)
then g(w) is a transition matrix for a PSL(N,C)-bundle but not for a SL(N,C)-bundle. This relation is similar to (2.4).
Let us choose a trivialization of E over D by choosing N linear independent holomorphic sections~s= (s1, s2, . . . , sN). Thereby, the bundle E over D is represented by a sum of N line bundlesL1⊕L2⊕· · ·⊕LN. The sections over Σg\w0are obtained by the action of the transition matrix~s0 =~sg.
Letm~ belongs to the root lattice (m~ = (m1, m2, . . . , mN)∈Q) (A.2). Transform the restric- tion of the section~son D×w0 as
sj →z−mjsj, j= 1, . . . , N. (2.14)
Then the transition matrix is transformed by the diagonal matrix g(z)→diag z−m1, z−m2, . . . , z−mN
g(z). (2.15)
It implies the transformation of line bundles over D Lj → Lj⊗ O(mj).
In this way we come to the new bundle ˜E (the modif ied bundle). It is defined by the new transition matrix (2.15). This transformation of the bundle E to ˜E (or more exactly the map of sheaves of its sections)
Γ(E)Ξ(−→m)~ Γ( ˜E), Ξ(m)~ ∼diag z−m1, z−m2, . . . , z−mN ,
is calledthe modification or the Hecke transformation of type m~ = (m1, m2, . . . , mN). In field- theoretical terms it corresponds to the t’Hooft operator,generating by monopoles (see below).
Let us relax the conditionPN
j=1mj = 0. Then the modification Ξ(m) changes the topology~ of E. We come to a nontrivial bundle of degree deg ( ˜E) = deg (E) +PN
j=1mj. In next section we illustrate this fact.
Now assume that m~ belongs to the weight lattice P (A.4). Then the modification Ξ(m)~ changes the characteristic class of a PGL(N,C)-bundle E. To prove it let us pass to the basis of the fundamental weights (A.3)
~ m=
N
X
j=1
mjej =
N−1
X
k=1
nk$k.
It follows from (A.3) thatmj and nk are related as m1= 1
N((N−1)n1+ (N −2)n2+· · ·+nN−1), m2= 1
N(−n1+ (N −2)n2+· · ·+nN−1),
· · · · mN = 1
N(−n1−2n2− · · · −(N−1)nN−1), nk =mk−mk+1. Rewrite the modification in the form of the product of the diagonal matrices
Ξ(~n)∼
N−1
Y
k=1
diag z−nk$k
. (2.16)
It follows from (A.3) that the monodromy of this matrix around the pointz= 0 is exp −2πi
N
N−1
X
k=1
knk
!
IdN. (2.17)
Therefore, the characteristic class of the adjoint bundle is unchanged if
N−1
X
k=1
knk =N
N−1
X
j=1
mj = 0, (modN).
In this case the weight vector m~ belongs to the root lattice Q. Otherwise, we come to the non-trivial monodromy (2.17). It is an obstruction to lift the PGL(N,C)-bundle to a SL(N,C)- bundle. This element can be identified with the monodromy (2.4) and in this way with an element from H2(Σ,ZN). As it was mentioned above, the modified bundle ˜E can be lifted to a GL(N,C) bundle. Let us act on the modified sections (2.14) by the scalar matrix
h=z
2πi N
N−1
P
k=1
knk
IdN.
It is a GL(N,C) gauge transformation. The monodromy of the new transition matrix is trivial.
Therefore, we come to the GL(N,C) bundle. The bundle is topologically nontrivial – it has degree
p=
N−1
X
k=1
knk =N
N−1
X
j=1
mj. (2.18)
It follows from (2.17) that the characteristic classξ and the degree pare related as ξ = exp2πi
N p.
The set of modifications that changes the degree on p is defined as solutions of (2.18) in inte- gers nk.
Assume that the bundleE is equipped with a holomorphic connection. OnD×w0 it takes the form (∂z+Az)dzand can be considered as an element of the affine Lie coalgebraglb∗(N,C)(Dw×0) The gauge transformation (2.15) acts on Aw acts as the coadjoint action
(Az)jkdz→ zmk−mj(Az)jk(1−δjk)−mjz−1δjk
dz. (2.19)
Let m~ ∈P. Then the first term in the r.h.s. is well defined, since mk−mj is integer. The last term represents the shift action (A.14) of the affine group ¯Wa (A.13) on the connection. The topology of E is not changed ifm~ ∈Qand we come to description of the characteristic class as elements from factor group ¯Wa/Wa (A.15). We come again to this point in Section 4.
Let N = pl with l 6= 1, N and Gl = SL(N,C)/Zl (A.9). Consider the gauge transforma- tion (2.16) with m~ ($)~ ∈ Γ(LG) (A.9). For example, we can take $~ = (p,0, . . . ,0). Then the monodromy (2.17) belongs to the group Zl. It means that the modified bundle ˜E is the Gl-bundle that cannot be lifted to the SL(N,C)-bundle (see (2.6)).
The modification can be performed in an arbitrary number of pointswa, (a= 1, . . . , n). To this end define the isomorphism classes of vector bundles as the quotient
n
Y
a=1
GL(N,C)(Dwa)\
n
Y
a=1
GL(N,C)(D×wa)/GL(N,C)(Σg\(w1, . . . , wa)).
We haventransition matricesga(za) representing an element of the quotient, wherezais a local coordinate. Let Ξ(m~a) denotes the modification of E at wa and Ξ = Qn
a=1Ξ(m~a). The order of modifications in the product is irrelevant, since they commute. To calculate the monodromy of Ξ we choose the same orientation in all points wa. The characteristic class of ξ of modified bundle ˜E corresponds to
n
Y
a=1
exp −2πi N
N−1
X
k=1
knak
! .
3 Bogomolny equation
Def inition. Let W =R×Σg. Consider a bundle V over W equipped with the curvature F. Let φbe a zero form onW taking value in sections of the adjoint bundle φ∈Ω0(W,EndV). It is the so-called Higgs field.
The Bogomolny equation onW takes the form
F =∗Dφ. (3.1)
Here ∗ is the Hodge operator on W with respect to the metric ds2 on W. In local coordinates (z,z) on Σ¯ g andyon the real line ds2 =g|dz|2+dy2, whereg(z,z)|dz|¯ 2 is a metric on Σg. Then the Hodge operator is defined as
?dy= 12igdz∧d¯z, ?dz =−idz∧dy, ?d¯z=id¯z∧dy, and (3.1) becomes
∂zAz¯−∂z¯Az+ [Az, Az¯] = ig(z,z)¯
2 (∂yφ+ [Ay, φ]), (3.2a)
∂yAz−∂zAy+ [Ay, Az] =i(∂zφ+ [Az, φ]), (3.2b)
∂yA¯z−∂z¯Ay+ [Ay, Az¯] =−i(∂z¯φ+ [Az¯, φ]). (3.2c) In what follows we will consider only PSL(N,C)-bundles.
A monopole solution of this equation is defined in the following way. Let ˜W = (W \~x0 = (y= 0, z=z0)). The Bianchi identityDF = 0 on ˜W implies thatφcan be identified with the Green function for the operator ?D ? D
?D ? Dφ=M δ(~x−~x0), (3.3)
M = diag(m1, m2, . . . , mN)∈gl(N,C), m~ = (m1, m2, . . . , mN)∈P (A.4), (3.4) and (m1, m2, . . . , mN) are the monopole charges. We explain below this choice of M. This equation means that φis singular at~x0.
Boundary conditions and gauge symmetry. In what follows except Section 3.1 we assume that∂yφvanishes wheny→ ±∞. It is the Neumann boundary conditions for the Higgs field, while the gauge fields are unspecified. LetV± be restrictions ofV to the bundles over Σg on the “left end” and “right end” ofW :y→ ±∞. These bundles are flat. It follows from (3.7a), where the gauge Ay = 0 is assumed. It was proved in [9] that in absence of the sourceM = 0 in (3.3) the only solutions of (3.1) with these boundary conditions areF = 0,φ= 0. Note that these boundary conditions differ from ones chosen in [9].
The Bogomolny equation defines a transformation V− → V+. (E and ˜E in our nota- tions in Introduction.) We will see in next sections that in general the characteristic classes of bundles are changed under these transformations. It depends on the monopole charges
~
m= (m1, m2, . . . , mN).
The system (3.2) is invariant with respect to the gauge groupG action:
Az →hAzh−1+∂zhh−1, Az¯→hA¯zh−1+∂z¯hh−1,
Ay →hAyh−1+∂yhh−1, φ→hφh−1, (3.5)
where h ∈ G is a smooth map W → GL(N,C). To preserve the r.h.s. in (3.3) it should satisfy the condition [h(~x0), M] = 0.
Assume for simplicity that V is an adjoint bundle. Since the gauge fields for y = ±∞
are unspecified and only flat we can act on them by boundary values of the gauge group G|y=±∞=G±. ThenM±={V±}/G± are the moduli spaces of flat bundles.
Relations to integrable systems. The moduli spaces of flat bundles are phase spaces of non-autonomous Hamiltonian systems related to the isomonodromy problems over Σg. The isomonodromy problem takes the form
[∂z+Az,Ψ] = 0, [∂z¯+Az¯,Ψ] = 0. (3.6)
Here Ψ ∈ Ω0(Σg,AutV) is the Baker–Akhiezer function. These system is compatible for any degree of bundle, because it is defined in the adjoint representation. One example of these systems we have mentioned in Introduction (V−→Painlev´e VI) and (V+→Zhukovsky–Volterra gyrostat).
It is known, that the moduli space of flat bundles are deformation (the Whitham deformation) of the phase spaces of the Hitchin integrable systems – the moduli spaces of the Higgs bundles. To consider this limit one should replace a holomorphic connection by the κ-connection κ∂z+Az
introduced by P. Deligne and take a limit κ → 0. It is a quasi-classical limit in the linear problem (3.6). Details can be found in [11,12,13,14]. In this way a monopole solution put in a correspondence (symplectic Hecke correspondence) two Hitchin systems (the first and the last examples in Introduction). But Bogomolny equation tells us more. It describes an evolution from one type of system to another.
It is possible to generalize (3.3) and consider multi-monopole sources P
aMaδ(~x−~x0a) in the r.h.s. This generalization will correspond to modifications in a few points of Σg described at the end of previous section.
It is interesting that in some particular cases this situation was discussed in the frame- works of a supersymmetric Yang–Mills theory [15,16]1. It was observed there that a monopole configuration corresponds to a soliton type evolution along y. Therefore, it can be suggested that the system (3.2) is integrable. We did not succeed to prove this fact, but propose a linear problem related to the Bogomolny equation. An associated linear problem allows one in princi- pal to apply the methods of the Inverse Scattering Problem or the Whitham approximation to find solutions [18]. Assume that the metricg on Σg is a constant. Then the system (3.2) is the compatibility condition for the linear system
∂z+Az+12λ−1g(∂y+Ay+iφ) ψ= 0,
∂z¯+Az¯+12λg(∂y+Ay−iφ) ψ= 0,
where λ ∈ CP1 is a spectral parameter. It can be suggested that monopole solution of (3.2) corresponds to a soliton solution of this system. We will not develop here this approach2.
Gauge f ixing. Choose a gauge fixing conditions as: Az¯ = 0. Holomorphic functions h=h(y, z) preserve this gauge. Then
−∂¯zAz = ig
2 (∂yφ+ [Ay, φ]),
1We are grateful to A. Gorsky who bring our attention to this point.
2The SU(2) case andW =R3 was analyzed in [19] for different boundary conditions.
∂yAz−∂zAy+ [Az, Ay] =i(∂zφ+ [Az, φ]),
∂z¯Ay =i∂z¯φ.
The last equation means that Ay −iφ is holomorphic. It follows from (3.5) that the gauge transformation of this function is
Ay −iφ→h(Ay−iφ)h−1+∂yhh−1.
Thus, we can keep Ay = iφ by using holomorphic and y-independent part of the gauge group (∂yh= 0). Finally, we come to the system
∂z¯Az =−ig
2∂yφ, (3.7a)
∂yAz−2i∂zφ+ 2i[Az, φ] = 0, (3.7b)
Ay =iφ, (3.7c)
Az¯= 0. (3.7d)
Two upper equations from (3.7) lead to the Laplace type equation
∂y2φ+ 4
g(∂z∂z¯φ+∂z¯[Az, φ]) = 0. (3.8)
In scalar case (3.8) is simplified
∂y2φ+ 4
g∂z∂z¯φ= 0. (3.9)
3.1 Rational solution in scalar case
In this subsection we replace Σg by C. The coordinates z,z¯ on C will play the role of local coordinates on Σg. Consider (3.9) on ˜W =R×C\(0,0,0). In this particular case we can choose the boundary conditions in the following form:
φ|y=±∞= 0, (3.10)
Az|y=±∞= 0. (3.11)
The solution of (3.9) withg= 1 satisfying (3.10) has the form:
φ=c 1
py2+z¯z, (3.12)
where cis a constant. So in fact we deal here with the Laplace equation on R×C\(0,0,0). It follows from (3.11) and from the equation∂z¯Az=−2i∂yφ (3.7a) that
Az(z,z, y) =¯ A+z(z,z, y),¯ y >0 and y= 0, z6= 0,
Az(z,z, y) =¯ A−z(z,z, y),¯ y <0, (3.13)
where
A+z(z,z, y) =¯ −ic 1 z
y
py2+zz¯− 1 z
!
+ const, A−z(z,z, y) =¯ −ic 1
z y
py2+zz¯+ 1 z
!
+ const,
and Az(z,z, y) is a connection on the line bundle¯ L over ˜W. The connection has a jump −2ic1z aty= 0. To deal with smooth connections we compensate it by a holomorphic gauge transform that locally near ~x0 has the form h ∼ zm. Here m should be integer, because h is a smooth function. Notice that all holomorphic line bundles over S2 are known to be O(m)-bundles, m∈Z. Thus, we have c=im2,m∈Z. This usually referred as a quantization of the monopole charge. In fact the constant c contains factor 4π (area of a unit sphere) which yields a proper normalization of delta-function and appears in Gauss’s law. The gauge transformation h is the modification (2.14), (2.15) for line bundles over CP1. This is what we mean saying that the described 3-dimensional construction characterizes the modification of the corresponding bundle.
Consider for a moment the general situation W = R×Σg and let z,z¯ be local coordinates on Σg. Locally near ~x0 = (0,0,0) connections corresponding to solutions of (3.9) have the form (3.13). LetS2 be a small sphere surrounding the point~x0 inW and Σg,± be the left and right boundaries of W and L± are the corresponding restrictions ofL. Then (as it is explained in [9] in detail)
Z
Σg,+
F = Z
Σg,−
F +m,
where F is a curvature of the connection A. In other words, the monopole solution with the charge mincreases the degree of bundle by m (degL+= degL−+m).
3.2 Elliptic solution in scalar case The Laplace equation (3.9) on Στ has the form
∂y2φ+ 4(Im(τ))2∂z∂z¯φ= 0, (3.14)
or
∂y2φ+ (2πα)2∂z∂¯zφ= 0, α−1 = 2πi τ−τ¯,
and Im(τ) is the area of parallelogram of periods. We give two representations of the Green function φ and prove their equivalence using the same technique as for the Kronecker series described in [17].
A naive elliptic solution of (3.14) on ˜W is obtained by averaging (3.12) over the lattice Γ =Z⊕τZ⊂C:3
φ(z, y) =cX
γ∈Γ
1
p(παy)2+|z+γ|2. (3.15)
However the series diverges. That is why we consider its generalization R(s, x, z, y) =cX
γ∈Γ
χ(γ, x)
((παy)2+|z+γ|2)s, R 12,0, z, y
=φ(z, y), (3.16)
where
χ(γ, x) =eα−1(γ¯x−¯γx)
3We omit here and in what follows the ¯z dependence.
is a character Z×Z → C∗ of the additive group Γ and s, x are complex parameters. The characters are double-periodic
χ(γ, x+ 1) =χ(γ, x), χ(γ, x+τ) =χ(γ, x), γ ∈Γ, while the series R(s, x, z, y) are quasi-periodic
R(s, x, z+ 1, y) =eα−1(x−¯x)R(s, x, z, y),
R(s, x, z+τ, y) =eα−1(x¯τ−¯xτ)R(s, x, z, y). (3.17) The variablexdescribes behavior ofR(s, x, z, y) on the lattice Γ. In other words,xparameterizes the moduli space of line bundles on Στ. Note that forRe s >1 the series in the r.h.s. of (3.16) converges. The function
R 12, x, z, y
=cX
γ∈Γ
χ(γ, x)
((παy)2+|z+γ|2)12 =φ(x, z, y) (3.18) is the formal solution of (3.14) with the quasi-periodicity conditions (3.17).
Another representation of the Green function can be obtained by the Fourier transform.
Define the delta-functions δ(y) =
+∞
Z
−∞
dpe2πipy, δ(2)(z,z) =¯ X
γ∈Γ
χ(γ, z).
Then X
γ∈Γ
χ(γ+x, z) =χ(x, z)X
γ∈Γ
χ(γ, z) =χ(x, z)δ2(z,z) =¯ δ2(z,z),¯
Let ˜φbe the Green function with the quasi-periodicity (3.17)
∂y2φ˜+ (2πα)2∂z∂¯zφ˜=cδ(y)δ(2)(z,z).¯ Expanding it in the Fourier harmonics we find
φ˜=− c 4π2
X
γ∈Γ +∞
Z
−∞
dp e2πipy
p2+|γ+z|2χ(γ+x, z).
Integrating over p provides factorπ and leads to the following expression:
φ(x, z, y) =˜ − c 4π
X
γ∈Γ
1
|γ+x|e−2π|γ+x||y|χ(γ+x, z). (3.19) It is worthwhile to note that the solution (3.19) is well defined. Our goal is to find interrelations between (3.19) and (3.18).
Consider a generalization of (3.19) I(s, x, z, y) = 2cπsys−12 X
γ∈Γ
Ks−1
2
(2π|y||γ+x|)
|γ+x|s−12 χ(γ+x, z). (3.20)
Here Kν is the Bessel–Macdonald function Kν(2πyz) = Γ(ν+12)(z)ν
2(πy)νΓ(12)
+∞
Z
−∞
dp e2πipy (p2+z2)ν+12
.
The function I(s, x, z, y) is the Green function for the pseudo-differential operator
∂y2+ 4α2π2∂z∂¯z
s
on R×Στ with the boundary conditions (3.17). Since K1
2(x) = r π
2xe−x,
we conclude that for s= 1 I coincides with ˜φ(3.19) up to constant.
We are going to establish a relation between (3.16) and (3.20), and in this way between (3.15) and (3.19). Let us prove that
I(s, x, z, y) =cX
γ∈Γ
∞
Z
−∞
dp
∞
Z
0
dt
t tse−t(p2+|γ+x|2)+2πipyχ(γ+x, z). (3.21) In fact, using the integral representation for the Gamma-function
Γ(s) =
∞
Z
0
dt
t tse−t (3.22)
and taking the integral overt in (3.21) we come to (3.20).
The representation (3.21) is universal and can serve to defineR(3.16)
Lemma 3.1. The function R(s, x, z, y) has a representation as the Fourier integral R(s, x, z, y) = 1
Γ(s)χ(γ, z) Z ∞
∞
dk I
s, z, x, k πα
e−2πiky. (3.23)
Proof . Substitute in (3.23) I(s, x, z, y) (3.21) and take first integral over k. We come to the condition p =παy. Then using the integral representation for the Gamma-function (3.22) we
obtain (3.16).
Remark 3.1. The series (3.20) is a three-dimensional generalization of the Kronecker series (see [17])
K(x, x0, s) =X
γ
χ(γ, x0)|x+γ|−2s.
Using the Poisson summation formula Kronecker proved that Γ(s)K(x, x0, s) =α1−2sΓ(1−s)K(x0, x,1−s)χ(x, x0).
Our purpose is to generalize this functional equation for the 3-dimensional case Στ ×R. It takes the following form.
Lemma 3.2. The function I(s, x, z, y) satisfies the functional equation:
I(s, x, z, y) =χ(x, z)π−12α−2s+1
+∞
Z
−∞
dk I 3
2 −s, z, x, k πα
e−2πiky. (3.24) Proof . Following [17] we subdivide integral (3.21) into two parts
I(s, x, z, y) =cX
γ∈Γ
∞
Z
−∞
dp
T
Z
0
dt
t tse−t(p2+|γ+x|2)+2πipyχ(γ+x, z) +cX
γ∈Γ
∞
Z
−∞
dp
∞
Z
T
dt
t tse−t(p2+|γ+x|2)+2πipyχ(γ+x, z), T ∈R, T >0.
The second term is a well defined function for alls. Consider the first one. It is well known that for the series
Θ(t, x, x0) =X
γ
e−t|x+γ|2χ(γ, x0) the following functional equation holds:
Θ(t, x, x0) = (αt)−1Θ α−2t−1, x0, x
χ(x0, x).
The latter follows from the Poisson summation formula which states that the averaging of function over some lattice equals the averaging of its Fourier transform over the dual lattice.
In the above case the functional equation appears after the Fourier transform for the Gauss integral. Then
X
γ∈Γ
∞
Z
−∞
dp
T
Z
0
dt
t tse−t(p2+|γ+x|2)+2πipyχ(γ+x, z)
=X
γ∈Γ
∞
Z
−∞
dp
T
Z
0
dt
t tse−tp2−α−2t−1|γ+z|2+2πipyχ(γ+z, x)χ(x, z)(αt)−1
integrating overp
= X
γ∈Γ T
Z
0
dt
t tse−π2y2t−1−α−2t−1|γ+z|2χ(γ+z, x)χ(x, z)(αt)−1 rπ
t
making substitution α−2t−1→t
= X
γ∈Γ
∞
Z
α−2T−1
dt t t32−s√
πα2−2se−t((παy)2+|γ+z|2)χ(γ+z, x)χ(x, z).
Let T =α−1. Then I(s, x, z, y) =cX
γ∈Γ
∞
Z
α−1
dt t t32−s√
πα2−2se−t((παy)2+|γ+z|2)χ(γ+z, x)χ(x, z)
+cX
γ∈Γ
∞
Z
−∞
dp
∞
Z
α−1
dt
t tse−t(p2+|γ+x|2)+2πipyχ(γ+x, z). (3.25) The proof follows from (3.25). One should only substitute I(s, x, z, y) from (3.25), into (3.24).
Formula (3.25) represents I as the sum of two terms. Direct evaluation shows that the first (of two) term from the l.h.s. of (3.24) equals to the second one from the r.h.s. and vice versa.
From Lemmas3.1 and 3.2we come to the main result of this section R(32 −s, x, z, y) =
√πα2s−1
Γ(32−s) I(s, x, z, y).
Now put s= 1. Then one can see that well-defined series πX
γ∈Γ
χ(γ+x, z)e−2π|y||γ+x|
|γ+x| (3.26)
describes the analytic continuation of the divergent series πX
γ∈Γ
χ(γ, x) 1
p(παy)2+|γ+z|2.
We use (3.19) as the Green function. Then Az(z,z, y, x) =¯ −ic
4π 1
π2α2sgn(y)X
γ∈Γ
1
γ+xe−2π|γ+x||y|
χ(γ+x, z), (3.27)
sgn(y) = 1 for y≥0, sgn(y) =−1 for y <0.
Notice that the jump of A (while coming through y = 0, z = 0) is obviously defined by the jump of sgn(y).
Remark 3.2. Note that (3.27) is a formal solution of the Bogomolny equation. Forx6= 0 it is not a connection of a line bundle over Στ due to its monodromies similar to (3.17). We will use this solution in next section to define a genuine connection for higher ranks bundles.
In order to compare elliptic configuration with the rational we takex= 0. Then on the line y= 0 the connection is proportional to
Az ∼X
γ6=0
1
γχ(γ, z) =E1(z)−α−1(z−z),¯
where E1(z) = ∂lnϑ(z) is the so-called first Eisenstein series and ϑ(z) is the theta-func- tion (2.10). E1(z) has a simple pole at z = 0 with Resz=0E1(z) = 1 and the connection Az
is double-periodic. In terms of (3.5) the gauge transformation h compensating the jump of the connection is given by an integer power of theta function ϑm(z), m∈Z. Thus
∂logh=∂logϑm(z) =mE1(z).
4 Arbitrary rank case
Here we describe modification of vector bundles of an arbitrary rank. First, we repeat arguments of [9] and justify the choiceM in (3.4). As before, we consider PSL(N,C) =Gad-bundles.
Near the singular point ~x0 the bundle V is splited in a sum of line bundles. Using the solution (3.12) for a line bundle we take the Higgs field near the singularity in the form
φ= i
2p
y2+z¯zdiag(m1, . . . , mN).
It follows from (3.13) thatAz undergoes a discontinuous jump at y= 0 A+z −A−z = i
zdiag(m1, m2, . . . , mN). (4.1)
To get rid of the singularity ofA atz= 0, as in the Abelian case, one can perform the singular gauge transform Ξ that behaves near z= 0 as (2.16)
Ξ = diag z−m1, z−m2, . . . , z−mN .
Assume thatm~ belongs to the weight latticem~ ∈P. It means that Ξ is inverse to the cocharac- ter γad of PSL(N,C) (γad ∈t(Gad) =P∨ ∼P (A.5)). As it was explained before, the modified bundle V+ can not be lifted to a SL(N,C) bundle. On the other hand, if m~ = (m1, . . . , mN) belongs to the root lattice Q (A.5), then Ξ−1 = ¯γ and there is no obstruction to lift V+ to an SL(N,C) bundle. Note that (4.1) describes the affine group ¯Wa (A.13) action in the former case and the affine group Wa (A.12) action in the latter case. From field-theoretical point of view it is an action of the t’Hooft operator on Az (see (2.19)).
IfN =pl,(l6= 1, N) one can consider the intermediate situation andγGl(A.8). It means that
~
m ∈ t(LGl) ∼Γ(Gp). This embedding provides the modification that allows the PGL(N,C)- bundle to lift to the Gl = SL(N,C)/Zl-bundle but not to a SL(N,C) bundle. In this way the monopole charges are related to the characteristic classes of bundles.
One can use the maps to the Cartan subgroups of the solutionφ(z) for a line bundle over Στ (3.26) withx= 0
φ→φ·diag(m1, m2, . . . , mN).
Unfortunately, in this case V being restricted on Στ is splitting globally over Στ and defines an unstable bundle, though it allows one to describe its modifications.
There exists a map ofφ(z, y, x) and Az(z, y, x) with x 6= 0 to a non-semisimple elements of sl(N,C)
0 k1φ(z, y, x1) . . . kN−1φ(z, y, xN−1)
0 0 . . . 0
... . . .. ...
0 . . . 0
,
0 k1Az(z, y, x1) . . . kN−1Az(z, y, xN−1)
0 0 . . . 0
... . . .. ...
0 . . . 0
.
Since these matrices commute they are solutions of the matrix equation (3.8). The connection has a jump aty = 0. The bundle is characterized by the diagonal monodromy matrices (2.7)
ρa= diag(a1, a2, . . . , aN), ρb = diag(b1, b2, . . . , bN), where
a1=
N−1
Y
j=1
σ
1 N
j , a2=a1σ−11 , aN =a1σ−1N−1, b1 =
N−1
Y
j=1
ς
1 N
j , b2=a1ς1−1, bN =a1ςN−1−1,
σj = exp(α−1(xj−x¯j)),ςj = exp(α−1(xjτ¯−x¯jτ)). Note that they arey-independent. Moreover, the singular gauge transform, leading to a continues solution of the Bogomolny equation, belongs to the upper nilpotent subgroup and in this way does not change the topological type of the bundle.
Now we describe non-diagonal modifications Ξ of a PGL(N,C)-bundles over Στ. We do not know solutions of the Bogomolny equation in this case, and only can assert that the modification
“kill the jump” ofAz aty= 0:
Ξ−1∂zΞ =A+z −Ξ−1A−zΞ.
We use the global description of a bundleE in terms of the transition matrices ρa,ρb (2.7) using the approach of [1]. Let
ρa= IdN, ρb=e−u, (u= diag(u1, u2, . . . , uN)), e(a) = exp(2πia). (4.2) The group commutator of these matrices is IdN. Thereby,E can be lifted to a SL(N,C)-bundle.
Define a modification Ξ of E to the bundle ˜E with the transition matrices (2.8). Then Ξ should intertwine the transition matrices
Ξ(z+ 1, τ) =Q ×Ξ(z, τ), (4.3)
Ξ(z+τ, τ) = Λ(z, τ)×Ξ(z, τ)×diag(e(u)). (4.4)
The matrix Ξ(z) degenerates atz= 0 and we assume that it has a simple pole. These conditions fix Ξ(z). It can be expressed in terms of the theta-functions with characteristics
Ξkj(z, u1, . . . , uN;τ) = θ
k
N −12
N 2
(z−N uj, N τ) θN1(z, τ) , where
θ a
b
(z, τ) =X
j∈Z
exp 2πi
(j+a)2τ
2 + (j+a)(z+b) .
The quasi-periodicity properties (4.3), (4.4) follow from the properties of the theta-functions θ
a b
(z+ 1, τ) =e(a)θ a
b
(z, τ), θ
a b
(z+a0τ, τ) =e
−a02τ
2 −a0(z+b)
θ
a+a0 b
(z, τ).
This modification has the type (N−1N ,−N1, . . . ,−N1). The modification that allows to lift ˜E to GL(N,C)- bundle is
Ξ1(z) =h(z)Ξ(z) =θ k
N −12
N 2
(z−N uj, N τ), where the gauge transformation h is the diagonal matrix
h(z) =θN1 (z, τ)IdN.
This modification intertwine the boundary conditions (4.2) with ρa=Q, ρb= ˜Λ, Λ =˜ e−2πi(Nz+2Nτ )Λ.
The last transformation belongs to GL(N,C). Moreover, it can be proved that det
Ξ1(z, u1, . . . , uN;τ) iη(τ)
= ϑ(z) iη(τ)
Y
1≤k<l≤N
ϑ(ul−uk) iη(τ) ,
where η(τ) =q241 Q
n>0(1−qn) is the Dedekind function (q = exp 2πiτ) andϑ(z) is the theta- function (2.10). Since ϑ(z) has a simple pole in Στ the bundle ˜E is a GL(N,C)-bundle of degree one. This modification provides the Symplectic Hecke correspondence between the elliptic Calogero–Moser system and the Elliptic Top.
Now consider the modification of the trivial bundle E with the transition matrices (4.2) to the ˜E=El(2.11), (2.12), whereup = (˜u1,u˜2, . . . ,u˜p) is the moduli of the modified bundle. The modification takes the form
Ξkj(z, τ) = θ
k
l −12
l 2
(z−l˜ui), lτ)
θ1l(z, τ) , (j=mp+i, m= 0, . . . , l−1).
As it was explained in Section2the modified bundle can be lifted toGl= SL(N,C)/Zl-bundle, but not to SL(N,C)-bundle.
A SL(N, C ) and PSL(N, C ) [20, 21]
The group SL(N,C) is an universal covering of PSL(N,C) with the centerZN =Z/NZ
Id→ZN →SL(N,C)→PSL(N,C)→Id. (A.1)
Thereforeπ1(PSL(N,C)) =ZN. The both groups have the same Lie algebraG.
Roots and weights. The Cartan subalgebraH⊂Gis a hyperplane inCN H=
x= (x1, . . . , xN)∈CN|
N
X
j=1
xj = 0
. The simple roots Π ={αk}
α1=e1−e2, . . . , αN−1=eN−1−eN
form a basis in the dual space H∗. Here {ej} j = 1, . . . , N is a canonical basis in CN. They generate the set of roots of type AN−1
R={(ej−ek), j 6=k}.
The root lattice Q⊂H∗ takes the form Q=nX
mjej|mj ∈Z, X
mj = 0o
. (A.2)
We identifyH∗ andH by means of the standard metric onCN. Then the coroot system R∨=
α∨(R) = 2(α∨, β)
(β, β) ∈Z for any β ∈R
coincides with R, and the coroot lattice Q∨ coincides with Q.
The fundamental weights $k, (k = 1, . . . , N −1) are dual to the basis of simple coroots Π∨∼Π ($k(α∨k) =δkj)
$j =e1+· · ·+ej− j N
N
X
l=1
el, (A.3)
$1=
N −1 N ,−1
N, . . . ,−1 N
, $2 =
N −2
N ,N −2
N , . . . ,−2 N
, . . . ,
$N−1 = 1
N, 1
N, . . . ,1−N N
.
In the basis of simple roots the fundamental weights are
$k= 1
N[(N−k)α1+ 2(N −k)α2+· · ·+ (k−1)(N −k)αk−1
+k(N −k)αk+k(N −k−1)αk+1+· · ·+kαN−1].
The fundamental weights generate the weights lattice P ⊂H∗, P =
( X
l
nl$l|nl ∈Z )
, (A.4)
P =
N
X
j=1
mjej, mj ∈ 1
NZ, mj−mk∈Z. The weight lattice is generated byQ and the vector
$1=e1− 1 N
N
X
j=1
ej.
The weight lattice P defines representations of SL(N,C), while Q define representations of PSL(N,C).
The factor-groupP∨/Q∨ (P∨ ∼P) is the centerZN of SL(N,C). On the other hand it can be identified with the cyclic group symmetryej →ej+1 mod(N) of the extended Dynkin graph Π∪(α0=eN −e1).
Characters and cocharacters. Let ¯T (Tad) be a Cartan torus in SL(N,C) (PSL(N,C)).
Define the groups of characters4
Γ =¯ {χ(x)}¯ ={T →¯ C∗}, Γad={χad(x)}={Tad →C∗}.
They can be identified with lattice groups in H∗ as follows. Let $k be a basic weight and φ= (φ1, φ2, . . . , φN), φk = 2πi1 lnxk. The functions
exp 2πi($kφ), k= 1, . . . , N −1 generate a basis in ¯Γ. Similarly, for αk∈Π
exp 2πi(αk, φ), k= 1, . . . , N−1 is a basis in Γad. Thereby, we have
Γ =¯ P, Γad =Q.
Define the dual groups of cocharacterst( ¯G) = ¯Γ∗ and t(Gad) = Γ∗ad as the maps t( ¯G) ={¯γ =C∗→T },¯ t(Gad) ={γad =C∗ → Tad}.
4The holomorphic maps of the tori toC∗such thatχ(xy) =χ(x)χ(y) forx, y∈ T.
